Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian

Ising spin Hamiltonians are often used to encode a computational problem in their ground states. Quantum Annealing (QA) computing searches for such a state by implementing a slow time-dependent evolution from an easy-to-prepare initial state to a low energy state of a target Ising Hamiltonian of quantum spins, HI. Here, we point to the existence of an analytical solution for such a problem for an arbitrary HI beyond the adiabatic limit for QA. This solution provides insights into the accuracy of nonadiabatic computations. Our QA protocol in the pseudo-adiabatic regime leads to a monotonic power-law suppression of nonadiabatic excitations with time T of QA, without any signature of a transition to a glass phase, which is usually characterized by a logarithmic energy relaxation. This behavior suggests that the energy relaxation can differ in classical and quantum spin glasses strongly, when it is assisted by external time-dependent fields. In specific cases of HI, the solution also shows a considerable quantum speedup in computations.

1) The authors make a comparison of residual energy in their protocol [claimed to be inversely proportional to annealing time in Eq. (25), see also Eq. (26)] with a classical behavior that is vanishing logarithmically as in Eq. (3).
I'm not sure if this is an "apple with apple" comparison.
Quite often, an additional requirement/assumption put on (quantum) annealing is that the problem Hamiltonian should be extensive in the number of qubits --and I'm wondering if there are no such assumptions behind Eq. (3)? How would the protocol described in the manuscript behave for extensive problem Hamiltonian? (e.g., with the expected scaling of DeltaE_I with N for such H_I) On the other hand, the authors focus primarily on "the most complex" problem Hamiltonian with exponentially many terms in the Ising formulation. The obtained scaling in Eq. (35) hinges on the behavior of the density of states possible for such a problem Hamiltonian. I'm wondering how the classical annealing would behave under such an assumption --is Eq. (3) still valid in that case?
I believe that the authors should take special care to be clear about such assumptions thought the paper.
2) Below Eq. (8), they cite [10] as a place where Hamiltonian in Eq. (8) also appeared. If I recall correctly, it was also considered in [13], though only concerning sufficient time for ground state preparation.
3) Below Eq. (22), the authors state that an exponentially large number of terms in H_I would add an extra exponential overhead on Monte-Carlo approaches. But wouldn't it also put a similar overhead on Quantum annealing? --somehow, all that exponentially many terms would have to be stored and imposed also for quantum protocol. Is it a fair comparison? 4) At the beginning of Sec 4.D.2, the authors talk about "small connectivity". Please be more specific, e.g., is the Edwards-Anderson model of small connectivity in that context. How about the Sherrington-Kirkpatrick model? Also, please add a citation.
In the inset of In the last paragraph of that subsection, the authors talk about "stiffest QA" (also in a couple of other places). It is jargon --please be more specific.

5)
There is a typo with the missing reference "[?]" below Eq. (20) and in a few subsequent places.
Reviewer #2 (Remarks to the Author): In this paper, the authors use an analytical solution, published in a number of previous publications, to argue that scaling of the residual energy with computation time in quantum annealing (QA) is polynomial, instead of logarithmic, e.g., Eq. (3), expected for classical annealing. If true, this is a significant result because it shows that QA is exponentially faster than its classical counterpart in reducing the residual energy. Unfortunately, I believe this result is obtained based on assumptions that may not be justified, as I will discuss below.
First of all, I believe Eq. (3) is obtained under the constraint that reaching the ground state at the end of computation is guaranteed. In practice, however, if the goal of computation is just to reduce the residual energy much shorter computation times may be sufficient.
Second, although the analytical solution described in Sec. II for the specific initial Hamiltonian of Eq. (8) is impressive, the final result summarized in the equation above (21) is what you expect for an unstructured quantum search algorithm without slowing down at the minimum gap location (locally adiabatic evolution). Generalization of this result to the more physically motivated Hamiltonian of Eq. (18) is rather handwavy but may still be okay, meaning that Eq. (23) may remain valid.
Finally, the main result of this paper that is summarized in Eq. (25) and (26) is obtained under the assumption that the energy levels of the Ising Hamiltonian are linearly spaced by an energy distance \delta = \Delta E_I/N, as discussed above Eq. (25). This assumption plays the key role in the conclusion that the residual energy is reduced polynomially instead of logarithmically with time. For most Hamiltonians, however, the density of states is expected to be maximally packed near the center of the spectrum with an exponential decrease as the ground state is approached. This can be shown easily for simple Hamiltonians such as Eq. (35), but may also be verified for more complex Hamiltonians. As a result, the residual energy will have logarithmic dependence on <n>, instead of linear as assumed. Therefore, the dependence of the residual energy on the computation time becomes logarithmic, similar to the classical case.
I therefore believe that the main conclusion of this paper is an artifact of an unjustified assumption. For this reason, I cannot accept this paper for publication.
Here are some more specific comments: -The index t of H_t in Eq. (1) can be confused with time.
-On page 1, the typical energy spacing is stated to be \Delta E_I/2^N, which is not true as discussed above.
-\beta in Eq. (3) which is also repeated in the abstract in undefined.

Previous equation
3. The last subsection in the section of Results is shortened in order to keep the length of the article within the Nature Communications limits.
4. Title is shortened to "Analytical solution for nonadiabatic quantum annealing to arbitrary Ising spin Hamiltonian".
5. Structural changes according to the Nature Communications formatting guidelines.
************************************************ We would like to thank both of the reviewers for their valuable comments and suggestions, which have helped us substantially improved the quality of the manuscript. Below is the point-to-point responses to the reviewers: Responses to Reviewer #1: Reviewer: I believe that this is an important contribution that should enjoy high recognition and open new directions in this field of quantum annealing. As such, I would like to recommend working towards its publication in Nature Communications. I, however, believe some sharpening of the presentation is necessary, please see the points below: Response: We are grateful to Reviewer 1 for very positive response. We address all Reviewer's suggestions/questions below.
Reviewer: 1) The authors make a comparison of residual energy in their protocol [claimed to be inversely proportional to annealing time in Eq. (25), see also Eq. (26)] with a classical behavior that is vanishing logarithmically as in Eq. (3). I'm not sure if this is an "apple with apple" comparison. Quite often, an additional requirement/assumption put on (quantum) annealing is that the problem Hamiltonian should be extensive in the number of qubits -and I'm wondering if there are no such assumptions behind Eq. (3)? How would the protocol described in the manuscript behave for extensive problem Hamiltonian? (e.g., with the expected scaling of E I with N for such H I ).
Response: The logarithmically slow relaxation in time is the feature of all systems that are called glasses. So, as a phenomenological consequence, a similar logarithmic dependence on the annealing time in Eq. 3 (now Eq. 22) has been expected to be true for all glass systems -not only extensive systems. Our solution addresses the systems of both type equally well. Hence the scaling laws that we found are equally applicable to both strongly connected and lattice-like spin systems.
1 Reviewer: On the other hand, the authors focus primarily on "the most complex" problem Hamiltonian with exponentially many terms in the Ising formulation. The obtained scaling in Eq. (35) hinges on the behavior of the density of states possible for such a problem Hamiltonian. I'm wondering how the classical annealing would behave under such an assumption -is Eq.
(3) still valid in that case? I believe that the authors should take special care to be clear about such assumptions thought the paper.
Response: We would like to respectfully disagree with Reviewer. The scaling in Eq. 26 (new Eq. 24) is found in our solution for all possible Ising Hamiltonians. Moreover, for energy relaxation it does not depend on the system size. So, these equations are equally applicable to lattice-like spin systems as to random spin clusters. Until the following subsection, our discussion was completely general, without assuming the limit of maximal complexity. However, we agree with Reviewer that the generality of Eqs. 25 for the residual energy (new Eq. 26) has to be discussed additionally. The resubmitted manuscript now explains that the typical systems that have spectra with a decreasing density of states when approaching the ground energy, do not show a logarithmic residual energy dependence on the QA time but they typically show a power-law relaxation, with exponential tail at the transition to the truly adiabatic regime.
Reviewer: 3) Below Eq. (22), the authors state that an exponentially large number of terms in H I would add an extra exponential overhead on Monte-Carlo approaches. But wouldn't it also put a similar overhead on Quantum annealing? -somehow, all that exponentially many terms would have to be stored and imposed also for quantum protocol. Is it a fair comparison?
Response: We agree with Reviewer that this was a point that needed more explanation. Hence, we added extra discussion to the main text. Rough estimates are as follows. If we have exponentially many, ⇠ 2 N , terms in the Hamiltonian, then to find only one eigenvalue we need ⇠ 2 N time. So the time to sort the array of eigenvalues takes 2 2N steps. On the other hand, for QA, it takes about 2 N steps to set couplings between the qubits, and then the annealing time ⇠ 2 N to perform QA. Hence, the net time still scales as ⇠ 2 N , i.e., QA is much faster.
Reviewer: 4) At the beginning of Sec 4.D.2, the authors talk about "small connectivity". Please be more specific, e.g., is the Edwards-Anderson model of small connectivity in that context. How about the Sherrington-Kirkpatrick model? Also, please add a citation. Response: We also would be interested. However, such questions are hard to answer because explicitly time-dependent simulations are very time-expensive even for relatively low (12 in our case) spins. We did statistical averaging for this figure using a LANL's supercomputer, and computation time is proportional to g. So, reaching a significantly larger g values without computation errors, unfortunately, is beyond our numerical capabilities. This is why the models like our's are needed -to make an insight into the numerically inaccessible regimes. On the other hand, in the revised manuscript we added a new section (the last section in Methods) which includes discussion of the scaling of the residual energy. A few more model studies are included which do not require brute force computations. This allows us to reach significantly large values of g. In these cases power-law scaling always holds.
Reviewer: In the last paragraph of that subsection, the authors talk about "sti↵est QA" (also in a couple of other places). It is jargon -please be more specific.
Response: We agree with Reviewer, in the new version we define the model with all nonzero random parameters as the "maximal complexity" case, and we avoided using "sti↵" terminology.
Reviewer: 5) There is a typo with the missing reference "[?]" below Eq. (20) and in a few subsequent places.
Response: The citation errors have been fixed.

Responses to Reviewer #2:
Reviewer: In this paper, the authors use an analytical solution, published in a number of previous publications, Response: Please let us clarify here. We were writing not one more article on some famous model. The solution of our model has been published by one of us in appendix to a paper in 2014 as an illustration of a certain mathematical concept. Since then, it has never been reproduced elsewhere and has never been applied to physics or information science. So, we hope Reviewer can see that identifying importance of this solvable model for quantum annealing is a nontrivial step on its own. In fact, it took some steps in our article to connect the solvable model in the previously known form to the considered quantum annealing problem. Moreover, only one formula for the final state probabilities was used in our present article. All other formulas that we wrote are not found in the original solution.
to argue that scaling of the residual energy with computation time in quantum annealing (QA) is polynomial, instead of logarithmic, e.g., Eq. (3), expected for classical annealing. If true, this is a significant result because it shows that QA is exponentially faster than its classical counterpart in reducing the residual energy. Unfortunately, I believe this result is obtained based on assumptions that may not be justified, as I will discuss below.
Response: As we will detail below, we would like to respectfully disagree with some of the Reviewer's arguments. However, we agree that they are worth a discussion, so we also made proper changes in the main text to avoid similar critics from the readers.
Reviewer: First of all, I believe Eq. (3) is obtained under the constraint that reaching the ground state at the end of computation is guaranteed. In practice, however, if the goal of computation is just to reduce the residual energy much shorter computation times may be su cient.
Response: We would agree with Reviewer in that one can derive a bound on the annealing schedule, ⇠ 1/ log T , where T is the classical annealing time, so that if the annealing is performed slower than this, the ground state is guaranteed to be reached. We also agree that at short time, the relaxation is not universal and can be much faster. Eq. 3 (now Eq. 22) has been always used to describe long-time tail of the relaxation and only in spin glasses. This is the regime in our focus.
However, the logarithmic ⇠ 1/ log T with > 1 scaling refers to the time-dependent relaxation of the residual energy for classical annealing in a very large class of glass systems. It is the scaling that is actually used to define a transition to the glass phase. So, if we believe in glasses, we believe in this logarithmic law. A more complex question is whether this law is expected for quantum annealing through a glass phase. We cited the literature that suggested the same logarithmic law for quantum annealing, with a di↵erent interpretation of T . In numerous prior numerical studies, such a logarithmic energy relaxation during finite time quantum annealing through a glass phase has always been confirmed. So, now it is generally accepted, and has been even supported by non-rigorous but analytical arguments.
It is fair to compare with our 1/T energy relaxation as our solution applies to arbitrary Ising spin glasses in the same context as the previously conjectured logarithmic law.
We cited only a few papers in the original version about the logarithmic law. This could be not enough. In the revision we have extended the bibliography to review this topic.
Reviewer: Second, although the analytical solution described in Sec. II for the specific initial Hamiltonian of Eq. (8) is impressive, the final result summarized in the equation above (21) is what you expect for an unstructured quantum search algorithm without slowing down at the minimum gap location (locally adiabatic evolution).
Response: Although our time estimates for a general H I may look not impressive, first, we note that it is better than the result expected from classical annealing of spin glasses. This proves that quantum annealing is free of certain drawbacks of classical annealing, which often performs worse than a brute-force random search. Second, we show that this result does not apply to the case with ground state degeneracy, for which even exponential speedup can be achieved to find a specific ground state. So, our findings are actually consistent with common belief that quantum algorithms cannot solve "typical" computational problems but can give strong boost in rare cases, e.g., by using collective e↵ects that are often provided by systems with strong energy degeneracy.
Reviewer: Generalization of this result to the more physically motivated Hamiltonian of Eq. (18) is rather handwavy but may still be okay, meaning that Eq. (23) may remain valid.
Response: We think that there is a misunderstanding here. We did compare our solvable protocol to the protocol with a decaying transverse field but we do not claim that the solvable result applies to the transverse field case. In fact, our conclusions based on this comparison are very di↵erent. On one hand, we show that the solvable protocol outperforms the transverse field in the maximal complexity limit and in situations with the ground state degeneracy. On the other hand, we show that the transverse field clearly outperforms the solvable one on small-connectivity spin glasses. So, our solution is not to apply to quantum annealing generally. Instead, it can be used to set certain limits on what is possible and disprove hypotheses such as the logarithmic energy relaxation in spin glasses.
Reviewer: Finally, the main result of this paper that is summarized in Eq. (25) and (26) is obtained under the assumption that the energy levels of the Ising Hamiltonian are linearly spaced by an energy distance = E I /N , as discussed above Eq. (25). This assumption plays the key role in the conclusion that the residual energy is reduced polynomially instead of logarithmically with time. For most Hamiltonians, however, the density of states is expected to be maximally packed near the center of the spectrum with an exponential decrease as the ground state is approached. This can be shown easily for simple Hamiltonians such as Eq. (35), but may also be verified for more complex Hamiltonians. As a result, the residual energy will have logarithmic dependence on hni, instead of linear as assumed. Therefore, the dependence of the residual energy on the computation time becomes logarithmic, similar to the classical case.
Response: First, we would like to respectfully disagree with Reviewer about the generality of Eq. 26 (Eq. 24 in the resubmitted version). It is for the scaling of the average number of excitations -not residual energy. Hence, this result is exact and valid for any energy dispersion.
For Eq. 25 (Eq. 26 in the new version), in the previous version we wrote that the deviations for the residual energy are expected but not of the logarithmic type. In the new version, we provide more evidence to this claim.
We agree with the referee that, near the ground state, an exponential tail of the density of states results in a logarithmic dependence of the energy E n to the level index n, e.g., E n ⇠ log n. However, hE n i ⌘ hlog ni 6 = loghni. In fact, our exact solution for the residual energy with an exponential spectral density of the form ⇢(E) = ae E near the ground state still reveals a power-law relaxation. (See the last section of Methods.) The average is very sensitive to the form of the probability distribution P n . To see this fact, consider a simple extreme example, where, beside the ground state with energy E 0 , all other states have the same energy E 1 . Hence the spectrum is sharply concentrated at the highest energy. In this case, the residual energy is where P 0 is given by Eq. (13) in the manuscript, hence, which means an exponential decay of the residual energy with the annealing time ⌧ a . Therefore, even though the exact form of the spectral density near the ground state becomes important, special care must be given when taking the average for the residual energy.
We note that the power-law scaling of the residual energy is valid for any power-law energy dispersion. Namely, for E n ⇠ n ↵ , we have " res ⇠ 1/g ↵ , where ↵ = 1 corresponds to the special case of uniform spectral density. This allows us to study the residual energy with general forms of the density of states close to the ground state. In the revised manuscript, we added a new subsection in the section of Methods with more comprehensive discussions. We have shown there that a power-law relaxation of the residual energy is generally expected. This is verified by our simulations including for both Gaussian and exponential density of states.
Reviewer: The index t of H t in Eq. (1) can be confused with time.
Response: We have changed the notation to H M with M refering to "mixing".